Problem: Let $f(x) = -3x^{2}+3x+3$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )?
Solution: The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-3x^{2}+3x+3 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -3, b = 3, c = 3$ $ x = \dfrac{-3 \pm \sqrt{3^{2} - 4 \cdot -3 \cdot 3}}{2 \cdot -3}$ $ x = \dfrac{-3 \pm \sqrt{45}}{-6}$ $ x = \dfrac{-3 \pm 3\sqrt{5}}{-6}$ $x =\dfrac{-1 \pm \sqrt{5}}{-2}$